She is recruiting random little folk to tame dragons. She doesn’t know these little folk. They could easily just take the dragon and do whatever they want, be it join the greens or fly away with it and just chill for a bit. There’s not really keeping them loyal to Rhaneyra.
Another issue is that she’s assuming the little folk that tame a dragon are willing to go to war. Not everyone is willing to face that level of danger. They may say they are and even mean it, but when the confrontation is about to happen and they realize the danger they’re in, they could easily renege.
Lastly, these people could be mad. They could potentially just burn a few important people when it’s least expected.
The whole strategy seems deeply flawed.
Because it is. Jace said it best if any lowborn bastard(s) can claim a dragon, how can he (and by what right) sit the Iron Throne after Rhaneyra is dead. This is desperate hour for Rhaneyra.